ON THE VIRASORO ALGEBRA AS REDUCED POISSON SUBMANIFOLD OF A KAC-MOODY ALGEBRA … · 2018-11-16 ·...

proceedings of theamerican mathematical societyVolume 122, Number 3, November 1994

ON THE VIRASORO ALGEBRAAS REDUCED POISSON SUBMANIFOLD

OF A KAC-MOODY ALGEBRA ON THE CIRCLE

GLORIA MARl BEFFA

(Communicated by Peter Li)

Abstract. In this paper we show how the Lie-Poisson structure on the dual of

the Virasoro algebra can be obtained through a standard Poisson reduction pro-

cess performed on a Kac-Moody algebra on the circle. We use the geometrical

idea of the process to establish some relation between transverse structures on

both Poisson manifolds.

Introduction

For a long time it has been known that the Korteweg-de Vries equation ut —

6uux - uxxx can be written as a Hamiltonian system with respect to two different

Poisson structures (see Lax's paper [9], or Adler's [1]). This bi-Hamiltonian

character gives rise to the definition of a sequence of Hamiltonian structuresand commuting Hamiltonian operators which can be generated by recursion

(see, for example, [2]). The second Hamiltonian structure in this hierarchycoincides with the canonical Lie-Poisson structure on the dual of the Virasoro

algebra. Later, in an attempt to generalize this property of the KdV equation,

Adler [1] defined a family of second Hamiltonian structures with respect towhich the generalized KdV equations could also be written as Hamiltonian

systems. Jacobi's identity for these brackets was proved by Gelfand and Dikiiin [5]. These Poisson structures are called second Hamiltonian KdV structures

or Gelfand-Dikii brackets, and they are defined on the space of «th-order scalar

differential operators (or Lax operators). In the case of second-order operatorswith a vanishing first-order term the Gelfand-Dikii bracket coincides with the

Lie-Poisson structure on the dual of the Virasoro algebra (it is also the only case

in which the second Hamiltonian KdV structure is linear).

Drinfeld and Sokolov established a connection between Lie algebras and the

Gelfand-Dikii brackets in their paper Lie algebras and equations of KdV type [4].

They showed how Gelfand-Dikii brackets could be obtained from the canonical

Lie-Poisson structure on a Kac-Moody algebra on the circle associated to a

semisimple Lie group G following certain reduction process. The case G =

SL(2, R) coincides with the Virasoro algebra.

Received by the editors October 18, 1992 and, in revised form, February 23, 1993.

1991 Mathematics Subject Classification. Primary 53C80; Secondary 53C57.

©1994 American Mathematical Society

0002-9939/94 $1.00+ $.25 per page

859

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

860 G. M. BEFFA

In this paper we offer a simpler explanation of their result for the Virasoro

algebra case using a standard geometrical reduction process. A similar idea hasalso been used by Casati and Pedroni in [3]. In their paper, Casati and Pedroni

show a rather algebraic way to obtain classical equations which are Hamiltonian

systems with respect to Gelfand-Dikii brackets, using Drinfeld and Sokolov

reduction. For that, they apply a modified theorem by Marsden and Ratiu

about Poisson reduction on bi-Hamiltonian manifolds. In this paper we also use

a theorem by Marsden and Ratiu which can be found in [ 10]. Although as stated

in [10] the theorem is valid for finite-dimensional Poisson manifolds and does

not include the case of the dual of the Virasoro algebra (a Fréchet manifold), a

minor argument would allow us to apply the theorem directly. Instead of doing

so, we prefer to repeat the geometric arguments in [10] so that the reader maybe made more familiar with the geometry of these manifolds. In this way, we

prove the existence of a reduced Poisson bracket on a certain quotient manifold.

Rather than showing how to obtain Hamiltonian equations on this quotient, we

show that, furthermore, the quotient manifold is Poisson-isomorphic to the dual

of the Virasoro algebra (Theorem 2.1). This procedure is geometrically clearer

than that in [4] since the algebra of pseudodifferential operators is not used.It is also a more appropriate approach to work in the geometrical propertiesof these brackets. For example, it allows the establishment of some relation

between transverse structures for the Lie-Poisson brackets on the dual of the

Kac-Moody algebra on the circle associated to SL(2, R) and on the dual of the

Virasoro algebra. We comment on this relation at the end of §2.

1. Kac-Moody algebras on the circle and their Poisson structuresinherited from the corresponding current algebra

Let G be a real semisimple Lie group and g its Lie algebra. Let G =C^iS1 ; G) be the group of smooth functions on the circle with values on G

(or group of loops) with the pointwise product. G is also called the current

group. Its Lie algebra g = C°°(Sl ; g) is called the current algebra, and it has

the pointwise Lie bracket as Lie algebra structure. Denote by B the Killing

form of the Lie algebra g, and define the following scalar product on g :

(L,M)= [ B{L(d),M(d))d6.Jo

This scalar product enables us to embed g in its dual space. Next, define

œ{L,M) = j\(L{8),d-^)d6.

It is easy to check that

(o([L, M],N) + œ([M, N],L) + œ([N, L],M) = 0

for all L, M, N e g . That is, co is a cocyclefor g (see [13]). Define the Liealgebra g = g © R with Lie bracket

[(L, s), (M, t)] = ([L, M], œ(L, M)).

g is called the central extension of g obtained through œ (see [13]). In this

particular case g is also called a Kac-Moody algebra on Sl .

Next we will state some of the known results about these structures. For their

proofs see, for example, Pressley and Segal [13] or Reiman and Semenov [14].


ON THE VIRASORO ALGEBRA 861

Proposition 1.1. With the appropriate identifications the coadjoint representation

of the algebra g is given by the formula

(1.1) ad*(M, t)(L, s) = ([M, L} + s^-, o\ ,

for any L, M e g and any t, s € R. The center of g acts trivially ong*,

and therefore the coadjoint action above can be derived from the action of G on

g*. This action is given by the formula

(1.2) Ad*(g)(L,s)=(AdG(g)(L) + s^g-i,s} ,

where Adg represents the adjoint action of G on g.

Because of (1.1) and (1.2), the space g* is usually identified with the space

of matrix differential operators of the form

(1.3) -SJ8+L^

where Leg* and sgR. Under this identification the coadjoint action is

simply the gauge action of G on the space of differential operators of the form

(1.3). The coadjoint orbits correspond to the symplectic leaves of the usual

Lie-Poisson structure of the algebra, which in this case is given by the formula

{X, 5f}(L, s) = (\d%f, d&], L) + sco(dJT, d5?)

•l ' d(d%f)-SAi:

+ [dJr,d5?],L) de(1-4) Jo V dd

• iB{ad*(dJr){d5f,s),L)dd.

>o

From (1.2), we observe that the central parameter s is fixed under the coad-

joint action of the group. Therefore, the Kac-Moody algebra g stratifies into

Poisson submanifolds corresponding to different values of the parameter; eachone of these submanifolds is isomorphic to g endowed with the Poisson bracket

( 1.4). We will focus our attention on the study of the hyperplane given by s = 1.

Let us denote it by g*. The Lie-Poisson structure on this hyperplane is referred

to as a second Hamiltonian structure on g*.From now on we will restrict ourselves to the case G = SL(2, R), although

we will frequently point out what to do in the general case.

Given an operator P = -^ + M(6) with M e C°°(Sl ; g), consider the

associated differential problem || = M¿¡. We will denote by T(P) the mon-

odromy matrix associated with ^| = MÇ, (that is, if X is a fundamental matrix

solution of || = MÇ, then by definition T(P) = X(0)X-1{2n)). The followingknown results give a brief description of the coadjoint orbits (Kirillov symplec-

tic leaves) of the Poisson structure (1.4).

Theorem 1.1. The SL(2, R)-conjugation class of the monodromy T(P) com-

pletely determines the orbit of P.

Corollary 1.1. If T(P) = /, where I is the identity matrix, then the zero matrix

is in the coadjoint orbit of P and the codimension of such an orbit is three. If


862 G. M. BEFFA

T(P) = -I then —fa + (° ~J/4) is in the coadjoint orbit of P = -j§ + M{8)and the codimension of such an orbit is also three. Furthermore, the two orbits

corresponding to the identity and to minus the identity are the only ones that

have codimension three. Any other orbit has codimension one.

Proof. The first part is an immediate consequence of Theorem 1.1. To prove

the second part we look at the expression of the coadjoint action (1.1) and

observe that the codimension of the orbit of P = —jg + M coincides with the

number of independent periodic solutions of equation ^ + [Y, M] = 0. Any

solution of this equation is of the form Y = X(6)XoX(6)~l, where Xo € g and

X is a fundamental matrix solution for j¡¡ = Mt¡. Therefore, Y is periodic ifand only if

Xo = T(P)X0T(P)-1.

Any matrix T(P) either fixes the whole algebra (namely, when T(P) — I or

T(P) — -I) or only a 1-dimensional subalgebra (which would be generated by

T(P) - trace(T(P))I). Therefore, the codimension of a symplectic leaf will bethree whenever the monodromy is ±1 and one in the remaining cases. □

In [14] it was shown that the conjugation class of T(P) is the only invariantunder the coadjoint action. Notice also that all the results described up to nowcan be formulated in exactly the same way if one replaces the group SL(2, R)by a general semisimple Lie group G.

To finish this section we will give an explicit formula in coordinates of the

bracket (1.4) which will be used in §2.

Definitions 1.1. Let X\, X2, Xj, be generators of sl(2, R) given by

MÍA)- MA*)- MÍO-On sl(2, R) consider the scalar product given by (X, Y) = traced, Y) and

let X*, Xj , X% be the dual elements for X\, X2, X^ with respect to ( , ).

With this notation, any element X e C°°(S{ ; sl(2, R)*) can be written asl =X\X* +x2X2 + x^X* for some 27r-periodic real-valued functions Xi, x2, X3.

Define e'k : g -► R to be the kth Fourier coefficient of the component x¡,for i = 1, 2, 3 ; that is,

4 (AT) = / * T¡-e-iketmx{XXi)dd,Jo 2n

i = 1,2,3 and k = ... , -2, -1, 1,2,.... The family {ek}k,i provides

a complete set of coordinates for g*, and the gradient of e'k is given by

fc-iMXi € g ■

In these coordinates it is straightforward to check that the Lie-Poisson bracket



( 1.4) looks like

{el, e2d}(X) = f ' * ¿-'(^[trace«*,, X2]X - idXxX2)]ddJo lZ7tl

■" 2lt^k+d{X)'

{el, ed}(X) = j * -i_e-^+rf)fl[trace([Jri, X3]X - idXxXz)]dd

2¿eLd(x)>

{e2 , e3d}(X) = [ * -±-e-^k+d^[tmce([X2, X3]X - idX2X3)] d8Jo lZ7tJ

2n'= -h^M{X)

and

{el, ed}(X) = -£ôl+d, {e¡ , e2}(X) = ^o¡+d, {e¡ , e¡}(X) = -£o¡+d,

where we have made use of the sl(2, R)-commutation relations [X\, X2] = X-¡,[Xi, X3] = X2, [X2, X3] = Xi and of the fact that trace(AriJT/) = 0 whenever

i ^ j and traced2) = -trace(Z|) = trace(X|) = \ and where ôj is the

Kronecker delta, that is, ôj = 0 if i ^ j and ôj = 1 for all i.

2. The Virasoro algebra as a Poisson submanifold of g*

In this section we will describe a certain foliation on a submanifold J? of

g*. This is in fact the canonical Drinfeld-Sokolov construction. This foliation

will be defined by an integrable subbundle of the tangent bundle of Jf, and its

corresponding space of leaves will be isomorphic to the dual of the Virasoro al-gebra. We will then check that the chosen subbundle satisfies enough conditionsso as to guarantee the existence of a Poisson bracket on the space of leaves. This

reduced bracket will be canonically induced by the Lie-Poisson bracket (1.4).

The induced structure will also coincide with the Poisson structure on the dual

of the Virasoro algebra.Consider Jf to be the subset of g* given by differential operators of the

form

(21) p = -± + (a ß

{A) dd + \l -a

In the general case of any semisimple Lie algebra g, one would consider a

standard gradation for g and define J? as the set of differential operators of the

form -^-rT+L(0), where T is a generator of the graded component of degree-1 and L is a matrix function which takes values on the Borel subalgebra of

g corresponding to elements of nonnegative grading. In the SL(2, R) case thelength of the grading is one, the negative graded part corresponds to strictlylower triangular matrices (with generator ( ° ^ ) ), and the Borel subalgebra is

given by the upper triangular matrices, resulting in expression (2.1) above.


864 G. M. BEFFA

The reason for restricting ourselves to the submanifold Jt is that an operatorof the form (2.1) is equivalent to the second-order scalar differential operator

d2 . , , , „ /, dL=^-2 + {a'-a2-ß) ('=;£)

in the sense that if £ is a solution of the equation LÇ = 0, then the vectorv — (£' + aÇ, c¡) is a solution of the differential equation

dv (a ß \

dd={l -a)V-

As it was shown in [4], the same assertion can be made for SL(«, R) andGL(n, R) where one finds an equivalence of elements of Jf with scalar differ-

ential operators of order n of the form

d" dn~l d+ w»-i^^rr + --- + "i^ + "o-dd" "-'¿0«-i ' ' "ld6

and where w„_i = 0 in the case of SL(n, R).

Therefore, if we wish to define an induced Poisson structure on the manifold

of «th-order scalar differential operators, we need to identify matrix differentialoperators of the form (2.1) which have the same scalar equivalent. Fortunately

such an identification is easily achieved by factoring through the gauge actionof certain subgroup of G, as next proposition shows.

Proposition 2.1 [4]. Let P\ and P2, respectively, be differential operators of the

formp _ día, M d (a2 ß2\Pl--dd + [l -aj> P2--dê+{l -a,)'

Assume that they are equivalent to scalar differential operators L\, L2, re-

spectively, in the sense mentioned above. Then L\ — L2 if and only if thereexists g e C°°(Sl ; SL(2, R)) of the form

(2.2) g(\ m\

with Ad*{g)(P2) = Px.

That is, the identification we are looking for is equivalent to factoring J?

through the gauge action of the (unipotent) subgroup of G consisting of ele-

ments of the form (2.2). We will call that subgroup N. In the general case one

factors by the gauge action of the unipotent subgroup which corresponds to the

Borel subalgebra that was initially chosen.

Proof. Assume Pi and P2 given as in the proposition and that g is given as

in (2.2), so that Ad*(g)(P2) - P\. From (1.2) we obtain that this condition isequivalent to

m = a\-a2, m'= ßi - ß2 + m(a2 + cn).

And from this relation we can deduce that

(2.3) a\ -a\-ß,=a'2-a\-ß2



so that L\ = L2 . And conversely, if L\ = L2 then (2.3) holds and for

one gets Ad*(g)(P2) = Px. D

Now that we know that the leaves of the foliation which we want to reduce

coincide with the coadjoint orbits under the action of Ñ, we will give some

conditions on the integrable subbundle of T(Jf) that defines this foliation so

that an induced Poisson structure can be canonically defined on the space of

leaves. These conditions are given by (i)-(iii) below, and they are analogous to

those stated in the theorem by Marsden and Ratiu in [10]. As we pointed out in

the introduction, with a little extra work we could apply the theorem directly,

but we prefer to repeat the argument to help the reader to understand better the

geometry of the manifolds involved.

Let JK be the subset of g* given by elements of the form (2.1). Let E be

the distribution generated by the Hamiltonian vector fields corresponding to thefunctionals {e\ + e|, for all k} . Then conditions (i)-(iii) for E are true.

(i) E is an involutive distribution since {e2. + e\ , ed + ed}(P) = 0.Besides, if we denote that by ô%? the gradient of the operator %f then,

using the definition of the Lie-Poisson bracket

{e\ + e\ , *}(P) = (oßT, ad*(¿(62 + e\))(P))

ddt

(ÔJr,Ad*(gt)(P))=j-t=o ai

(ß?oAd*(gt))(P),t=0

where X e C°°(Sl ; g*), P eJf and gt is a curve on Ñ such that ^|r=o =

¿(e2. + e|). The tangent space to the unipotent subgroup N is contained in

the subalgebra generated by ô(e\ + &\), and, therefore the integral curves of E

generate the coadjoint orbits of the action of N on J[.

Hence, the foliation defined by E n T(J?) is regular and Jf ¡Ñ coincideswith the space of leaves.

(ii) The second condition is that if K and L are smooth operators suchthat their differentials vanish on E, then the differential of the bracket {K, L}

should also vanish on E. That is, E should leave { , } invariant. This

property trivially holds for E since it is generated by Hamiltonian vector fields

(if {/, ft} = {/, g} = 0, then {/, {ft, g}} - 0 by Jacobi's identity).(iii) Finally, if we want the reduced bracket to be well defined we need the

additional property

{E±,-}(P)C TP(Jt) + EP,

for any Pe/. A short inspection using the definition of the bracket showsthat the vector space

{E1-, .}(/>) = {{X, -}(P) :{ßr,e2+ e¡}U - 0, for all k}

is generated by the Hamiltonian vector fields corresponding to ed + e\ for all

d, and, therefore, it coincides with EP .

Now, for any smooth operators / and ft on ^/N define the bracket givenby the formula

(2.4) {F,H}oi = {f,h} pon,


866 G. M. BEFFA

where i: J£ -» g* is the inclusion, n: Jt -> J!'/N is the natural projection,

and K and // are any smooth extensions of f o n and hon, respectively,

with differential vanishing on E.

Theorem 2.1. The bracket (2.4) is well defined and it induces a Poisson structure

in the space of leaves Jt/Ñ. Furthermore, the space (JT/Ñ, { , }Jt/p) is

Poisson isomorphic to a hyperplane of the dual of the Virasoro algebra with itscanonical Lie-Poisson structure.

Before going into the proof of the theorem, we will recall the definition of the

Virasoro algebra and the Lie-Poisson bracket on its dual and set some notation.Let vect^S1) be the algebra of vector fields on the unit circle, with the usual

commutator of vector fields as Lie bracket. On vect(5'1) define the so-called

Gelfand-Fuks cocycle

e(<»-»£)-jfíV"-The central extension obtained from vect(S') and c is called the Virasoro

algebra, and we will denote it by go • Its dual can be viewed as the set

2

g* = {{p(6)dd2, s), p(6) a 271-periodic function, s eR} = f^)51eR,

where <3>25' represents the 2-tensor algebra on the circle. For more details seeWitten [16] or Kirillov [8]. The Lie-Poisson structure on g¿* is, by definition,

(2.5)

{%•, 5f}o(p ,s)= f \ô<rô£" - sjr's&)p(6) dd + s f* ô%"ô&" de,Jo Jo

where here oß? represents the gradient of %? on the ^-component. As it hap-pened with the Kac-Moody algebra (and due to the central extension), g¿* strat-

ifies into Poisson submanifolds corresponding to a fixed choice of the centralparameter 5 ; all of them are geometrically equivalent except for the case s = 0.The hyperplane we are going to choose corresponds to the choice s = -1. Fi-

nally, if we denote by e^p) the kxh. Fourier coefficient of p , the bracket (2.5)is written in e-coordinates as

{e* , ed}o{p) = ^(k - d)ek+d - ^k3ô^+d .

Proof of Theorem 2.1. Conditions (i)-(iii) are going to guarantee that the bracket

(2.4) is well defined. In fact, let F be an extension of fon, and let H and H

be two different extensions of hon . Assume that the differentials of F, H, and

H vanish on E. Then dF e EL , and it follows from {F, •} e T(^) + E that

{F, H} o i is constant on the leaves of the distribution E. Moreover, H - H

vanishes on J! and the differential vanishes on E; {F, H - H}Poi = 0, and

therefore the definition does not depend on the extension we choose.Next, remember that each orbit can be identified with a periodic function

p{6) = a' - ß - a2 for some a and ß such that P = -j§ + {" Ia) belongs

to the orbit. This relation establishes an isomorphism between the space of

leaves and C°°-functions on the circle. We will view the latter manifold as the

underlying manifold for the hyperplane s = -I in the dual of the Virasoro



algebra, and we will show that their Poisson structures coincide. Fix {e^}, the

kth Fourier coefficients, as coordinates on Jf ¡Ñ.

Auxiliary Lemma 1. Let P = -j$ + (" _fQ), and let ek be the kth Fourier

coefficient of a' - ß - a2. Then Tk given by

T*(P) = ikel(P) - £ elp(P)el_p(P) + e2(P) - e¡(P)p

are extensions of ek o n for all k. Furthermore, their differentials vanish on Ealong Jf.

Proof of Lemma l.LetI = (;/J. If we identify sl(2, R) with its dualthrough the Killing form on the algebra, the basis dual to {X\ ,X2,X{\ isgiven by X\ = 2XX, X\ = -2X2, and X¡ = 2X3. Therefore, X can bewritten as

X = 2aJT, + ß(X2 + X3) + X3-X2 = aX* + ±(1 - ß)x; + \{ß + 1)*3*

so that we get the relations ek{a) = elk(X), ek(ß) = e^(Ar)-e^(Ar), and el(X) +

e2k(X) = ôl, for any Iê/.It is now clear that

rk(P) = ikel(P) - £ ep(P)el_p(P) + e2(P) - e¡(P)p

are extensions for ek o n on g*. Besides, e2. + e| and Td commute along Jt

for all k and d, and therefore the differentials of Tk all vanish on E alongJf. D

In order to finish with the proof of the theorem we need to show that thecommutation relations for Tk are the same ones that we had above for the

Virasoro algebra.

Following the definition (2.4), if L e J!/N and n(P) = L, we can showstraightforward that { , }J(. ~ is given by

i^,ed}^~(L) = {rk,rd}(P)

= l{d-k){i{k + d)E^ + g2+rf _ £3+rf]

(2.6) -l^2ir £ 4(pH+d-P(P) = [- 5> - p)4(P)4+d-P(P).

Auxiliary Lemma 2.

i(d - k)

p p

for all PeJ?.

Proof of Lemma 2. Assume P =—¡¡g + (* *a) £ ■* ■ Then elk(P) is the fcthFourier coefficient for the 27c-periodic function a. It is easy to check that (2.6)

can be obtained from the fact that the (k + d)th Fourier coefficients of ^- and

2a fß are equal. □


868 G. M. BEFFA

With the proof of this lemma we have obtained the following expression for

the reduced bracket:

to ' e¿Lr/*(L) = '( 2n ek+d(L) + Wáfc0+d

which coincides with the Virasoro algebra structure. (Note: this structure is

actually the negative of the Virasoro algebra structure. To have the exact

coincidence we should have chosen e¿ to be the kth Fourier coefficient ofß + a2 - a'.) D

Finally, some comments on transverse structures in infinite dimensions. The

concept of transverse structure was initially introduced by Weinstein in [15] for

finite-dimensional Poisson manifolds. The idea was to describe how the dif-ferent symplectic structures induced on the coadjoint orbits are "put together"

around a point which is singular for the Poisson tensor. One would choose an

appropriate submanifold transverse to the symplectic leaves and would study

(if possible) the Poisson structure induced canonically on it by the bracket of

the manifold. It was proved in [15] that all transverse directions in a finite-

dimensional manifold are equivalent in the sense that one can find an isomor-

phism of the Poisson manifold taking the induced structure on one of the trans-

verse submanifolds into the other one.It is not hard to prove that, if we choose a linear section transverse to the

orbit of maximal codimension, then a transverse structure for the bracket on g*

is given by the Lie-Poisson structure of sl(2, R)*. Besides, in [6] it was proved

that the submanifold Jf is transverse to the coadjoint orbits, that is, symplecticleaves of both Poisson manifolds have the same codimension. In [11] we chose

a linear section for the Virasoro algebra, transverse to the coadjoint orbit of the

point d2/dd2 + n2/2. The transverse structure found in [11] for the Virasoro

algebra bracket was not linear, though linearizable and therefore equivalent tothe Lie-Poisson bracket of sl(2, R)*. The reduction process is nonlinear in

nature (due to the presence of a2 in the identification leaves <-» C°°-functions

on the circle). This implies that linear sections project onto nonlinear sectionsof the quotient manifold and vice versa. It is clear to us that this is a causeof the nonlinearity of the transverse structure in [11]. Notice also that, with

the exception of G = SL(2, R) which corresponds to the Virasoro algebra, the

second Gelfand-Dikii is always nonlinear. The fact that for the Virasoro algebra

the final bracket is linear is certainly surprising.

Several interesting problems remain to be solved. It is reasonable to think

that in the general case of the Second KdV Structure associated with a semisim-

ple Lie group G, directions which are transversal to the leaves with / and -/

monodromy can be found such that the transverse structure along them is equiv-

alent to the Lie-Poisson structure on g*. This can probably be solved using the

definition (2.4) in the general case. It would also be very interesting to learn,

using the geometry of these algebras, how the Poisson bracket behaves when we

cross transversally coadjoint orbits which do not have maximal codimension.

Acknowledgments

The author thanks Professor Jack F. Conn for introducing her to the subjectof Poisson geometry and infinite-dimensional Lie algebras and offers special

thanks to Professor Peter Olver for his support.



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Department of Mathematics, University of California at Davis, Davis, California

95616Current address : Department of Mathematics, University of Wisconsin, Madison, Wisconsin

53706E-mail address : maribef f Qmath. wise. edu


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